Basically, paper towels win out over regular air dryers, jet air dryers, and cloth rollers, at least in settings like health care provision where hygiene is especially important. But here's a sketch of the arguments,based on a review of 12 studies on hand-drying since 1970. Summary statements are mine: quotations are from the study. As usual, footnotes and citations are omitted for readability.

Removing water from hands after hand-washing is an important part of killing the germs.

"For centuries, hand washing has been considered the most important measure to reduce the burden of health care–associated infection. ... Although studies have reported the importance of thorough hand drying after washing, the role of hand drying has not been widely promoted, and its relevance to hand hygiene and infection control seems to have been overlooked. Lack of attention to this aspect may negate the benefits of careful hand washing in health care."

Paper towels are the most hygienic of the hand-drying options: they dry skin faster, help remove contamination through friction, and don't risk spreading germs through the air.

"Although jet air dryers had drying efficiency similar to paper towels, their hygiene performance was
still worse than paper towels. The differences in bacterial numbers
after drying with air dryers and paper towels could be due to other
factors rather than the percentage of dryness alone. Friction can
dislodge microorganisms from the skin surface during both hand washing
and drying. Antimicrobial agents in soaps have too little contact time
to have bactericidal effects during a single use or with sporadic
washings, making friction the most important element in hand drying. It
is likely that paper towels work better because they physically remove
bacteria from the hands, whereas hot air dryers and jet air dryers
cannot. In many instances, however, rubbing hands with hot air dryers to
hasten drying would only lead to greater bacterial numbers and airborne
dissemination. It might be that rubbing hands causes bacteria to
migrate from the hair follicles to the skin surface. Many studies have
found friction to be a key component in hand drying for removing
contamination. ..."

"Hot air dryers are generally not recommended for use in health care settings because such dryers are relatively slow and noisy and their hygiene performance is questionable. Cloth roller towels are not recommended because they can become common use towels at the end of the roll and can be a source of pathogen transfer to clean hands. Recently, jet air dryers have undergone independent certification within the food safety arena in Australia, attesting to their increased hygiene benefits as opposed to the traditional hot air-drying method. However, the criteria and process of obtaining this type of certification remain questionable. The health and safety aspects of jet air dryers for use in locations where hygiene is paramount should still be carefully examined by the scientific community. Therefore, this makes paper towel drying, during which little air movement is generated, the most hygienic option of hand-drying methods in health care."

Air dryers, and especially jet dryers, are noisier. They can irritate skin.

"Air dryers, particularly jet air dryers, are obviously noisier than paper towels or cloth towels. ... [T]he mean decibel level of using a jet air dryer at 0.5 m was 94 dB, which is in excess of that of a heavy truck passing 3 m away. When 2 jet air dryers were used at the same time, the decibel level at a distance of 2 m was 92 dB. Therefore, in washrooms with jet air dryers, the noise level could constitute a potential risk to those exposed to it for long periods. ... "

"Use of air dryers may cause hands to become excessively dry, rough, and red. ... Affected persons often experience a feeling of dryness or burning; skin that feels rough; and erythema, scaling, or fissures. When the hands become irritated, health care workers may not wash their hands as often or as well. Concern regarding this effect of air dryers could become an important cause of poor acceptance of hand hygiene practices."

The environmental effect of paper towels is slightly worse than air dryers, but only very slightly.

"Using paper towels is more costly than using air dryers. Paper towels must be replaced frequently, whereas air dryers usually require little maintenance. ... However, air dryers can be costly to purchase and install. Therefore, those responsible for facility management should perform a careful cost analysis to determine whether they are cost-effective in their building."

People prefer to use paper towels--and people's preferences have value in this overall calculation, too.

"Another survey of 2516 US adults in 2009 still found that most people preferred to dry their hands with paper towels. If they had a choice, 55% of respondents selected paper towels, 25% selected jet air dryers, 16% selected hot air dryers, 1% selected cloth roller towels, and 3% were not sure. ... Hence, given the strong preference for using paper towels, hand hygiene adherence would possibly decrease if paper towels are not available in washrooms."

As the conclusion of academic studies often love to point out, there are vast possibilities for future research on this topic that go beyond the questions already discussed.

"Does the quality of paper towel have an effect on hand hygiene adherence? When recycled paper is used for hand drying, what kinds of studies are appropriate to assess the cost benefit of using recycled paper? Many questions remain unanswered. ... The maintenance of a clean environment around paper towels is also important. Paper towels deposited in bins could act as a bacteriologic reservoir if disposal is not managed properly. ... The risk of potential contamination among dispenser exits, paper towels, and hands should be considered in the design, construction, and use of paper towel dispensers. Architects working in the health care industry should also be aware of these issues when designing equipment for new facilities."